Features for flag-type power generation

ABSTRACT

Flag-type wind and/or water power generation features are described. Fluid-filled embodiments are described. The fluid may be employed for pumping effect and/or dynamic physical property change. Leading-edge and trailing-edge mounted flag architectures are described with various applications including flapping flag arrays and Vertical Axis Wind Turbines (VAWTs).

RELATED APPLICATIONS

This filing claims the benefit of and priority to U.S. ProvisionalPatent Application Ser. Nos. 61/718,096, filed Oct. 24, 2012,61/725,809, filed Nov. 13, 2012, and 61/739,975, filed Dec. 20, 2012,each of which are incorporated by reference herein in their entirety forall purposes.

FIELD

The embodiments described herein optionally relate to wind and/or waterpower generation, particularly electrical power generation.

BACKGROUND

Existing flag-type wind power generation devices are often illustratedas bio-inspired creations. One example of a fluttering “Piezo-tree”employs numerous tabs or flags (or so-called “leafs”), each attached toa PVDF piezo-based stem (or so-called “stalk”) with each such stem/stalkthen attached to a support structure. Reciprocal motion of the leavesand stalks driven by vortex shedding behind a bluff body mount producesAC electrical energy. The AC energy is then rectified to DC for storage.See, Li S., Lipson H., (2009), “Vertical-Stalk Flapping-Leaf GeneratorFor Parallel Wind Energy Harvesting,” Proceedings of the ASME/AIAA 2009Conference on Smart Materials, Adaptive Structures and IntelligentSystems, SMASIS 2009; Li, S., Yuan, J., and Lipson, H. (2011), “Ambientwind energy harvesting using cross-flow fluttering,” Journal of AppliedPhysics, 109, 026104. Another example of a sinuously-moving electric eelis described in Taylor et al., “The Energy Harvesting Eel: A SmallSubsurface Ocean/River Power Generator,” IEEE; Journal of OceanicEngineering (2001) 3; and Deniz Tolga Akcabay and Yin Lu Young,“Hydroelasic response and energy harvesing potential of flexiblepiezoelectric beams in viscous flow,” Physics of Fluids (2012).

Low conversion efficiencies due to material limitations persist in thesepiezo-based systems. Likewise, the need for rectifying AC energy(generated by flag oscillation) to DC energy for storage is alsoinefficient.

More generally, existing flag-type power generation approaches employ astandard or general (trailing) flag setup that only flutters at highervelocities. As such, power cannot be extracted from lower velocity flow.

Therefore, need for improvement exists on a number of fronts. Theembodiments described below variously meet these needs and/or others asmay be appreciated by those with ordinary skill in the art.

SUMMARY

In a first flag-type wind power generation architecture, flags areprovided with fluid-inclusion means. These means may comprise channelsor other configurations as elaborated upon below. In any case, the fluidis contained within a circuit typically including one-way checkvalve(s). Likewise, the flags are typically self-supporting such thatthey are able (at least in flow) to hold or support their internalchamber(s) up against the pull of gravity.

In use, motion of the flag in response to viscous flow (e.g., wind orwater) pumps fluid within the circuit. Included in a matrix or array ofdevices, the pumped fluid may be employed to run a generator forelectrical power output or maintained for use as mechanical power(converted by means of hydraulics for a given task or directlyemployed).

The flags in such devices or systems may be employed in a trailing-edgeor a leading-edge secured configuration, or at any angle of attack withrespect to the free stream between these bounding configurations. With atrade-off of increased design and/or system complexity, the former classof devices (optionally referred to as “inverted” flag designs) may offersignificant advantages in terms of cycle amplitude and/or low-flowapplications for their motion as compared to so-called standard orgeneral (i.e., leading-edge secured) flag configurations.

A second type of flag-type wind power generation architecturespecifically leverages the inverted flag approach with associatedadvantages. In these embodiments, flag sheets, strips, or panels(alternatively referred to as turbine “blades”) are employed in aVertical Axis Wind Turbine (VAWT) arrangement that presents each bladeto the wind in an inverted flag configuration at some period of timeduring each rotational cycle. With appropriate flexibility, the invertedflag reaches a critical point in flapping prior to the general flag,creating a boost in force on only one side of the turbine helping drivesteady state turbine rotation and/or startup. Following such flapping,the turbine blades reconfigure and create an increase in projected areaof the blade on one side of the turbine as compared to theother—again—helping drive steady state turbine rotation.

Each of the above designs are studied by various approaches in an effortto characterize their parameters and behavior. In connection with asimplified beam model, relevant quantities of inverted flag design arecharacterized in terms of non-dimensional bending stiffness β; in theVAWT example, a related metric, Cauchy number (Ca), is used and employedin a derivation from an existing analytical model to offer comparativeestimates of performance. In essence, β and Ca represent similarcharacteristics but for omission of Poisson's ratio in the case of Ca.Both 13 and Ca represent a correlation of the material and geometricproperties of a particular blade, along with the properties of thesurrounding fluid, and hence provide useful alternative system andstructural characterizations.

Irrespective of how they are analyzed, described, and/or represented,the flag structures incorporated in the flapping and/or turbine-typearchitectures may comprise hydro-skeleton constructions offering morecontrol over external forces (e.g., drag) acting thereon. Suchhydro-skeleton constructions are structures with an internal chamber.The chamber can be pressurized through a variety of fluids (e.g., air,water, oil, etc.). By changing the pressure of the chamber, thestructure will either bend (in the case with dissimilar materialproperties) and/or change the acting compliance (when the surroundingstructure is formed by a compliant material). Such approaches offeruseful tools in tuning the physical properties of flag elements in powergeneration devices, as well as other applications.

BRIEF DESCRIPTION OF THE DRAWINGS

The figures diagrammatically illustrate various example embodiments.Variations other than those shown in the figures are contemplated asdescribed in a broader sense per the Summary above, as genericallyclaimed, or otherwise.

FIGS. 1A-1D illustrate fluid-filled flag embodiment configurations.

FIG. 2 shows a power generation system example including the FIG. 1Bembodiment.

FIG. 3A schematically illustrates an example system for pressure headmeasurement including the FIG. 1B embodiment; FIG. 3B illustrates anexample wind-tunnel setup for the setup in FIG. 3A.

FIG. 4 schematically illustrates another example setup for testingfluid-filled flag embodiment performance employing a stepper motor togenerate oscillations as shown in FIG. 5A and a pressure trace as shownin FIG. 5B.

FIGS. 6A and 6B are side and top views, respectively, of a schematic foran example embodiment of an elastic sheet secured in a cantileverfashion.

FIG. 7A plots the Peak-to-peak amplitude A/L of a flag tip and FIG. 7Bthe Strouhal number fA/U as a function of bending stiffness β for massratio of O(1) for H/L=1.3 (squares), H/L=1.1 (circles) and H/L=1.0(triangles) where in FIG. 7B only the flapping mode with a constantflapping frequency was considered.

FIG. 8 illustrates superimposed flag sheets at four bending stiffnessvalues: (i) β=0.58 (U=2.8 ms⁻¹), (ii) β=0.26 (U=4.2 ms⁻¹), (iii) β=0.10(U=6.7 ms⁻¹) and (iv) β=0.06 (U=8.5 ms⁻¹) where μ=2.9 and H/L=1.1 withGravity into plane of the drawing in their different modes movement.

FIG. 9 plots the time history of the y-coordinate of the flag tipy(s=L)/L, β=0.26 (solid) and β=0.10 (dashed) where T is a flappingperiod.

FIG. 10A illustrates modes as in FIG. 8 for several cases of invertedflags (polycarbonate flag of thickness 0.008 m and length 30 cm) ofheight (H) ranging from 1 to 20 inches and wind speed ranging from 2.5to 6.5 ms⁻¹ where red is in a stable “straight” condition, green isflapping and blue is a re-stabilized deflected position. FIG. 10B chartsthe wind speed at which a given inverted flag starts flapping (i.e., theflow transition speed between the red and green regions as illustratedin FIG. 10A) as angle of attack to the wind changes.

FIG. 11A plots the hysteresis of amplitude A/L in mode transitionshowing increasing free-stream velocity (solid) and decreasingfree-stream velocity (dashed) where U* represents non-dimensionalfree-stream velocity and FIG. 11B plots first mode shapes of y/L and θobtained by proper orthogonal decomposition with β=0.26 (solid) andβ=0.10 (dashed) and fundamental mode shape of a linearizedEuler-Bernoulli beam (dashed-dot) where θ is the angle between the flagsheet and the −x axis in FIG. 6A/6B and μ=2.9 and H/L=1.1.

FIG. 12A plots Peak-to-peak amplitude A/L of the flag tip and FIG. 12Bplots Strouhal number f A/U as a function of bending stiffness 13 formass ratio of O(10⁻³) for H/L=2.0 (squares), H/L=1.6 (circles) andH/L=1.3 (triangles) where in FIG. 12B only the flapping mode wasconsidered.

FIG. 13A plots drag coefficient (C_(D)), FIG. 13B plots elastic strainenergy (E_(S)) and FIG. 13C plots the conversion ratio of fluid kineticenergy to strain energy (R) with mean C_(D) and E_(S) (circle) inbending and flapping mode and maximum C_(D) and E_(S) (square only forthe flapping mode, all for mass ration O(1).

FIG. 14 illustrates a sequential vortex formation process (i) to (vi),vorticity contour (β=0.19, μ=0.006) with flag sheet (solid line) atmaximum bending at (i) and (v).

FIG. 15 plots Formation Number (F) of the flapping mode for mass ratioO(1) (circle) mass ratio O(10⁻³) (square) where the Formation Numbersfor both positive and negative y-sides are averaged.

FIG. 16A is a top view that schematically illustrates an exampleembodiment of a wind turbine with a flag-type turbine design; FIG. 16Bis an annotated version of FIG. 16A.

FIG. 17 illustrates a side view of the turbine configuration of FIGS.16A and 16B wherein the dashed lines indicate wind tunnel boundaryconditions (floor and ceiling) for experimentation as described herein.

FIG. 18 illustrates the subject regimes of flexible flag turbine bladebehavior with shaded regions corresponding to shaded regions in FIG. 23.(Note that delineation between regimes does not represent preciseboundaries between regimes as boundaries vary in location depending onwind velocity and blade properties.)

FIG. 19A plots average torque in static tests of the example turbine ofFIGS. 16A and 16B with rigid flag blades at the pictured positions for 5ms⁻¹ (squares), 5.9 ms⁻¹ (circles), 6.9 ms⁻¹ (stars), 7.5 ms⁻¹(diamonds), and 8.5 ms⁻¹ (triangles) wind tunnel speeds. FIG. 19B plotsaverage torque in testing at the wind tunnel speeds for 1/32 inch thickDELRIN flag blades (squares and sum as dashed-dotted line), 1/16 inchthick DELRIN inverted flag blades (diamonds and sum as dotted line),rubber flag blades (circles and sum solid line), and rigid blades(triangles and sum as dashed line).

FIG. 20A plots instantaneous efficiency as a function of angularvelocity and FIG. 20B plots the efficiency for one rotation of theturbine as a function of tip-speed ratio for a turbine with the 1/32inch DELRIN blades.

FIGS. 21A and 21B plot power traces from two different dynamic testloading configurations, each over one period of rotation.

FIGS. 22A-22H are photographs of one rotation period for a dynamic testcase;

FIG. 23A plots power repeated rotation of the pictured setup and FIG.23B details the highlighted portion in FIG. 23A in reference to thepictured states in FIGS. 22A-22H.

FIG. 24 is a flowchart illustrating the cycle of output during oneturbine rotation in which inverted flag flapping is experienced.

FIGS. 25A and 25B plot test cases for rigid aluminum and highly-flexiblerubber flag blades, respectively (see legends), comparing analytical andtest results for torque through one half cycle of turbine rotation.

FIGS. 26A-26C schematically illustrates flag blade deformation predictedby an analytical model for the rubber blades; FIGS. 27A-27C arephotographs showing actual deflection under static testing conditions.

FIGS. 28A-28F are photographs demonstrating bending of a flag-type sheetor blade due to internal pressure manipulation.

DETAILED DESCRIPTION

Various example embodiments are described below. Reference is made tothese examples in a non-limiting sense, as it should be noted that theyare provided to illustrate more broadly applicable aspects of thedevices, systems, and methods described herein. Various changes may bemade to these embodiments and equivalents may be substituted withoutdeparting from their true spirit and scope. In addition, manymodifications may be made to adapt a particular situation, material,composition of matter, process, process act(s) or step(s) to theobjective(s), spirit or scope of the present subject matter. All suchmodifications are intended to be within the scope of the claims madeherein.

Fluidic Power Generation Flag

Flapping flag instability has been studied at length. As noted above,several groups have attempted to use this instability to create powerwith piezoelectric materials. The present embodiments utilize the sameinstability, but for pumping to generate fluid flow that can then beused to generate power.

A flag with an internal chamber that is placed in a wind stream flaps.While the flag is flapping, it acts as a pump. Through pumping fluid,associated generators are able to convert the fluid flow to electricity.In this way, a simple flag can be imbedded with a chamber and throughflapping is able to produce power. Flags can be used in both air and/orwater to produce power. The critical factor is having a fluid (i.e., gasor liquid) flow across the flag such that the flapping instabilityoccurs.

The subject flag can be implemented with chambers that have variousgeometries. As shown in FIG. 1A, a flag 10 includes a (relatively) largereservoir 12 connected to input and output tubes or channels 14, 16(respectively). Flag 20 in FIG. 1B includes a series of tubularswitchbacks 22. Either type of construction may be referred to as a“chamber” as it is optionally formed as negative or open space betweenopposing sheet layers 2, 4. The shape and location of the chamber can beotherwise optimized for performance.

Such variability in location includes the placement between the two walllayers (i.e., the chamber(s) may be closer to one wall than the other orthe walls on either side of the chamber may have different thicknesses),in which case the neutral axis of the chamber(s) is/are offset.Considered as a cantilever beam, once forces are introduced to differentpoints within the beam, it bends preferably towards one side. Suchintroduction may be in the form of fluid pressure as described furtherbelow.

Also (as alluded to above), configurations can be constructed withsingle or multiple chambers. Flag examples 30 with chamber 32 in FIG. 1Cand flag 40 with chambers 42 and 44 in FIG. 1D represent examples ofdesigns constructed for experiments to that effect. Different flagmaterials and pumping fluids can be used to improve performance undervarying fluid-flow conditions as well.

For a system 50 as illustrated in FIG. 2, in addition to flag format andmaterial (i.e., flag, outside flow, internal fluid), other propertiesselected to improve performance (e.g., one-way external check valves 52,54) can be used to force the uni-directional flow for pumping of theinternal fluid. Also, improvements to the uni-directional flow can alsobe provided through the shaping of the internal chamber geometries(e.g., by incorporating one-way valves like those found in biologicalvasculature). Offset or off-axis placement (e.g., as described above byconstructing the flag(s) with different layer material thickness) of thechambers or channels may also be employed to enhance fluid pumping.

Weight from the additional fluid within the internal chambers can aid inreaching the instability threshold that defines when flapping commences.Flag materials can be chosen that enhance the flapping behavior, whichin turn will enhance the pumping behavior within the internal chambers.Likewise, the flags can be mounted either behind a “flag pole” (in aso-called conventional, standard or general arrangement) or in front ofthe flag pole, which can enhance pumping as described below.

Conversion from mechanical to electrical energy can be done eitherthrough an embedded or an external converter. An external converter orgenerator 60 is illustrated in FIG. 2 with input and output lines 62, 64connected with valves 52, 63. Additional input and output lines 62′, 64′may be used in employing an array of flags 10, 10′, etc. connected in (afluidic) parallel arrangement. In addition, it is possible to create ahybrid generator with the addition of photovoltaic overlay layer(s) 66,photoelectric printing employing the flag material as a substrate, byother means, or as a part of the turbine set up.

As shown in FIG. 3A, to initially study the architecture, experimentslooked at a flag 10 (or 20, 30, etc.) constructed from fusedthermoplastic layers (in this case, PDMS material) defining an enclosedchamber and attached to a bar 70 up-stream of air flow (arrows) in awind tunnel. FIG. 3A also illustrates how an array of such devices maybe setup side-by-side (see dashed-line examples 10′, 70′). Alternativelyor additionally, multiple flag constructs 10, 10′ may be aligned orstacked in an array vertically along the same pole 70, 70′.

For study, the ends of an internal chamber (e.g., chamber 12 with inputand output channels 14, 16) were attached via check valves 52, 54 tovertical tubing 72, 74 as illustrated in FIG. 3B. The chamber withattached tubing was primed with colored water using a syringe. This wasdone such that the water level was the same in both tubes prior toinsertion in wind tunnel flow. With air flow in the tunnel then set to aspeed sufficiently high to produce flapping, a pressure head head(P_(head)) difference was observed as shown.

The flag constructions shown in FIGS. 1A-1C were tested. While designedto have chamber geometries of identical volumes, the observed pressurehead varied greatly between geometries per the Table 1 below:

Flag # Pressure Head (psi) 10 1.75 20 0.45 30 0.5Thus, the geometry of flag 10 can be considered the most effective atpumping fluid. However, other geometries can also be highly effective.

In another experiment, flags were mounted to a stepper motor. A flag 80was mounted to a motor shaft 82 to be driven through a specified angle(θ) to bounds 84, 84′ over a defined period (T) as illustrated in FIG.4. Through the flapping process, the internal pressure of the flagchamber(s) fluctuated. A position trace 86 shown in FIG. 5A was takenwhere flag 80 was constructed of rigid material and included a chamberwith a 2 mm offset from the center axis of the flag. As can be seen inFIG. 5B, the resulting pressure oscillation 88 matches the motion of thestepper motor providing further evidence of the manner in which pumpingenergy may be extracted from flapping flags in practical fashion,especially in connection with check valves employed to create one-wayflow with a consistently positive pressure profile.

Free Leading-Edge and Clamped Trailing Edge Architectures

In the various known methods of extracting energy from flapping flaginstability, third-party designs have varied many relevant parametersincluding the shape of the pole on which the foil is mounted, theweight, size, length, geometry of the foil itself, the media (rivers,oceans, wind) in which the foil is mounted and so forth. However, suchexamples all involve the situation where the foil is mounted behind a“flag pole” or other mounting device. In other words, the leading edgeof the flag has been fixed.

In contrast, certain embodiments hereof fix the trailing edge of theflag, leaving the leading edge free. There are several variations ofset-ups that will allow this orientation to function. Large amplitudeflapping motion has been demonstrated by the inventors hereof bymounting a thin plastic flag to a rigid stand. The magnitude of themotion demonstrated in suitable setups (as further characterized herein)can produce a higher efficiency conversion of energy from the freestream flow (wind, river, ocean current, etc.) to the mechanicalflapping. Also supporting this proposition is that the reversal of thefixed (i.e., clamped or otherwise secured) edge from leading edge totrailing edge allows for flapping at lower fluid flow speeds and, thus,a broader range of applicability. And even at higher speeds, in someinstances, the inverted flag may be configured to bend around its mount(180 degrees) and then operate similar to a standard or general flaggenerator design in which it may flap and continue to produce power.

Inverted flag geometry in power generation can be employed in a numberof ways. A first example is in systems as described in the previoussection (i.e., as a fluid-pumping flag optionally set in an array) givenparameters set forth below. Another example is in a vertical axis windturbine configuration as detailed in the next section.

Depending on the situation, the flag itself can be rigid or compliant.Tuning maybe accomplished using supplemental torsional spring(s), leafspring(s), or variation in the material from which the flags areconstructed. Also, the selected approach may change the preferred methodof energy conversion. Methods of energy conversion include the use ofpiezoelectric materials, use of an electromagnetic stator/rotorgenerator, and/or the fluidic approach described above.

The dynamics of an inverted flag (i.e., a flag with a fixed trailingedge) were investigated experimentally in order to characterize theconditions under which self-excited flapping can occur. The flappingdynamics were observed as closely related with periodic formation andshedding of leading-edge vortices.

Inverted flag behavior can be classified into three regimes based on anon-dimensional bending stiffness scaled by flow velocity and flaglength. Two quasi-steady regimes were observed (a straight mode, and afully deflected mode) in addition to a limit-cycle flapping mode withlarge amplitude that appears between the two quasi-steady regimes.Bi-table states are found in both a straight-to-flapping mode transitionand flapping-to-deflected mode transition.

Instead of maintaining a deflected shape at equilibrium, the invertedflag can flap with a large amplitude and store large strain energybecause of unsteady drag force. Indeed, a flapping flag sheet canproduce elastic strain energy that is several times larger than a sheetin the deformed mode, thereby improving the conversion of fluid kineticenergy to elastic strain energy. Moreover, (quite unlike the instabilityof the general flag) the effect of mass ratio relative to the magnitudeof flag inertia and fluid inertia on the non-dimensional bendingstiffness range for flag flapping is negligible.

In other words, an inverted flag structure as illustrated in FIGS. 6Aand 6B can be configured with high instability to low critical flowvelocity while providing high excitation amplitude. Therefore, it offersan excellent opportunity for successful application of flow-inducedoscillation to energy harvesting.

As studied in connection with FIGS. 6A and 6B, the dashed linerepresents the initial state or shape of a flag sheet 90, whichoscillates to the curved (solid line) positions. A is the distancebetween two amplitude peaks of tip 92 in the y-direction and is acurvilinear coordinate from the rear end mount 100 where flag 90 issecured as a cantilever beam when flag 90 is clamped or otherwise fixedto prevent rotation at 100.

In this example, the flapping dynamics of an inverted elastic flag sheet90 were investigated in order to characterize how its stability isinfluenced by parameters such as bending stiffness, flow velocity, fluiddensity, and sheet length. The effect of the parameters on drag andelastic strain energy was also examined. In addition, the flow structuredeveloped by the sheet was identified, and its relationship withflapping dynamics and strain energy conversion is discussed below.

Experiments were conducted in an open-loop wind tunnel capable ofproducing free-stream velocity U between 2.2 and 8.5 ms⁻¹. As in FIG.6A/6B, the downstream edge 94 of flag sheet 90 was clamped verticallybetween two long aluminum strips 2.5 cm wide and 1.3 cm thick. Theplates were made from polycarbonate (Young's modulus E=2.38×10⁹ Nm⁻²,Poisson's ratio v=0.38, density ρ_(s)=1.2×103 kgm⁻³) and sheet thickness(h) was 0.8 mm. The height of the sheet H was fixed at 30 cm, and thelengths of the sheet L were 23 cm, 27 cm and 30 cm, thereby providingaspect ratios H/L between 1.0 and 1.3. The deformation of the sheetoccurs primarily in the xy-plane, and the twisting of the sheet due togravitation was hardly observed. Thus, the deformation was largelytwo-dimensional. A small tip deflection 0.02<Δ/L<0.04 was observed inthe initial sheet configuration due to material defect.

For the observation of sheet motion, white plastic tape was attachedalong the top edge of the sheet, and its motion was captured by a highspeed camera (Nanosense MK3, Dantec Dynamics) mounted over the top ofthe test section. For each sheet, images were recorded at 100 frames persecond as wind speed was increased from 2.2 ms⁻¹ to 8.5 ms⁻¹. The topedge in the images was detected with a MATLAB script (Mathworks, Inc.).Aerodynamic drag D acting on the sheet was measured with two load cells(MBS, Interface, Inc.) connected to the top and bottom of the testsection. The drag was also measured with the clamping vertical stripsalone and subtracted from the total drag in order to obtain the net dragon the sheet.

As observed, two non-dimensional dynamical parameters are important forthe study of interaction between a fluid flow and an elastic sheet.These are the non-dimensional bending stiffness β and mass ratio μdefined below (Connell & Yue 2007; Alben & Shelley 2008; Michelin et al.2008):

$\begin{matrix}{{\beta = \frac{B}{\rho_{f}U^{2}L^{3}}}{and}{{\mu = \frac{\rho_{s}h}{\rho_{f}L}},}} & (1)\end{matrix}$

where B is the flexural rigidity of the sheet (B=Eh³/12(1−v²)), ρ_(f) isfluid density and ρ_(s) is sheet density. The value of β characterizesthe relative magnitude of the bending force to the fluid inertial force,and that of μ describes the relative magnitude of solid to fluidinertial forces. In the wind tunnel experiments, β ranged from 0.04 to1.50, and μ ranged from 2.5 to 3.3.

In order to investigate the effect of mass ratio on flapping dynamics,experiments were also conducted in water for low mass ratio of O(10⁻³).Flag sheets were clamped vertically in a free-surface water tunnel witha test section 1.0 m wide and 0.5 m high and a camera (IGV-B1920,Imperx, Inc.) was mounted below the floor of the test section, withimages of the bottom edge of the sheet were recorded at 10 frames persecond as the water speed increased. The water velocity ranged between0.15 and 0.53 ms⁻¹. Polycarbonate flag sheets at 0.8 mm thick weretested at 15 cm, 19 cm, and 23 cm lengths. While β ranged from 0.05 to1.38, μ was several orders of magnitude lower than that of the windtunnel experiments, ranging from 0.004 to 0.006.

In addition to capturing images of the sheet, planar digital particleimage velocimetry was performed in the water tunnel to visualizevortical structures of the flapping flag sheet. For this purpose, thetunnel was seeded with silver-coated hollow ceramic spheres of 70 μm(AG-SL150-16-TRD, Potters Industries). The particles were illuminated byan Nd:YAG laser sheet (Gemini PIV, New Wave) at the middle height of theflag sheet. Image pairs were captured at a rate of 15 pairs per second,and processed with PIVview (PIVTEC GmbH). Each pair of the images wascross-correlated with a multi-grid interrogation scheme. The firstinterrogation window size was 128×128 pixels with a 50% overlap, and thefinal window size was 32×32 pixels with a 50% overlap, which produces119×66 grids with the size of 5.8 mm.

As for results of such testing, amplitude and flapping frequency of theelastic flag sheets are presented for high mass ratio of O(1). Theresponses of the sheet are divided largely into three modes, dependingon non-dimensional bending stiffness β. FIGS. 7A and 7B plot and FIG. 8illustrates these results.

For β higher than 0.3, the flag sheet remained in a straight (mode (i),FIG. 8). For β lower than 0.1, the flag sheet bent in one direction andmaintained a highly curved shape (mode (iv), FIG. 8). Even though thesheet fluttered slightly in both Straight mode (i) and Deflected mode(ii), the peak-to-peak amplitude of the tip A/L is less than 0.2 inthese two modes and flutter periodicity was not clear. Between these twoquasi-static modes (0.1<β<0.3), the flag sheet flapped side to side, andthe deflection of the sheet was periodic with nearly constant A/L (modes(ii), (iii), FIG. 8). A/L increased drastically in the flapping mode,and plateaued to a range between 1.7 to 1.8. In 0.1<β<0.2, the sheetcontinued to bend past when the tip is at maximum |y|, which resulted inslight decrease in the |y|-position of the tip at maximum deformation asdescribed in FIG. 8 mode (iii) and FIG. 9. Within the flapping regime,unlike A/L (see FIG. 7A), the Strouhal number did not show a plateau buthas a maximum value of 0.14 around β=0.2 (see FIG. 7B).

Thus, inverted flag designs show several characteristics distinct fromthose of a general or standard flag with a clamped or pinned leadingedge and a free trailing edge. An inverted flag is able exhibit largerpeak-to-peak amplitude (e.g., up to A/L=1.7 to 1.8 or about 2) than thegeneral flag. The large amplitude is realized because the aerodynamicforce on the sheet.

Lift or drag destabilize the sheet from the initial “straight” position.Lift and drag act to destabilize the flag when moving—as shown in FIG.10A for several cases of flags of a height (H) ranging from 1 to 20inches and wind speed ranging from 2.5 to 6.5 ms⁻¹ from the red (higherbeta) to the green (mid range beta, where flapping is demonstrated).Drag increases with increasing velocity. At some point, this acts to“restabilize” the flag sheet in some sort of reconfigured shape as shownin the blue region in FIG. 10A.

While FIG. 10A considers the flags set parallel to the flow, FIG. 10Bgraphs the wind speed at which a flag starts flapping (i.e., thetransition speed between the red and green regions shown in FIG. 10A)considering angle of attack to the wind. In FIG. 10B, a clamp angle ofzero corresponds to the flag being aligned parallel to the flow. Becausedrag effects are more important when the flag is not perfectly parallelto the free stream than when the flag is parallel, a few things happen.First, the flag starts flapping earlier—however this flapping is of asmaller amplitude that the phenomenon seen when there is no angle ofattack. In order to look at this more fully the percentage ofamplitude/maximum amplitude flapping was changed at which the flag isconsidered to have fully reached the flapping state. This is what thethird axis (A/Amax) in the figure shows. So if the flag is flapping at30% (a value from which energy can be extracted therefrom) then if theflag is clamped at an angle it may flap sooner than it would otherwise(i.e., up to a point). Second, if the angle of attack is too large, theflag will not flap and will bend back (similar to the last row in FIG.10A when the flag goes from the red to the blue configurationsimmediately without transitioning through the large amplitude flappingregion). Third, if full flapping is considered to be very largeamplitude (e.g., 90% of max) it takes a higher flow velocity than thatof the flag parallel to the fluid flow.

Generally speaking, one will find that an inverted flag will be able toflap at lower velocities than a flag in a general or standardarrangement. However as can be seen that it is possible (depending onthe flag type and geometry) to completely miss entering an invertedflapping region. In other words, while the general flag exhibitsperiodic or chaotic flapping motion beyond a single critical bendingstiffness, the inverted flag experiences large-amplitude oscillationonly within a specific range of the bending stiffness.

On the other hand, an inverted flag with very small bending stiffnessbeyond the bending stiffness range studied here may behave differently.The very flexible sheet may bend around the clamped trailing edge andbecome parallel to the free stream, which results in the configurationsimilar to the general flag.

Subcritical bifurcation and bistable states are found in the invertedflag (see FIG. 11A). Bistable states exist in both straight-flappingmode transition and flapping-deflected mode transition. In increasingnon-dimensional free-stream velocity U*=√1/β=U√μ_(f)L³/B, the criticalvelocities U*_(c) are 2.1 in the straight-flapping mode bifurcation and3.4 in the flapping-deflected mode bifurcation for the sheet of μ=2.9and H/L=1.1. Then, when U* decreases from the deflected mode, the flagsheet tends to maintain its deformed shape and eventually has a slightlylower U* (=3.3) than U*_(c) of the increasing velocity. From theflapping to the straight mode, the initial condition of periodicflapping causes U*_(c) (=2.0) to be also lower than U*_(c) of theincreasing velocity. The U* widths of the two hysteresis loops are 0.1,and the corresponding dimensional wind velocity ranges in the hysteresisloops are about 0.3 ms⁻¹. In spite of small width of the bistableregion, the hysteresis was exhibited consistently in both transitionsfor every test.

Mode shapes of the flapping sheet were obtained by Proper OrthogonalDecomposition (POD) with the time series of the flag position data(Berkooz, Holmes & Lumley 1993). The y-coordinate of the sheet isdecomposed with orthogonal modes φ_(k)(s) as y(s, t)=Σa_(k)(t)φ_(k)(s)in the curvilinear coordinate s and time t. The first POD mode φ₁(s)with the largest eigenvalue is dominant at which the amplitude of thesheet increases monotonically from the root to the tip (FIG. 11B). Theroot mean square error of the y-coordinate approximated only by thefirst mode was within 1.5% of the flag sheet length over a cycle; theerror increased generally as the sheet underwent large deformation suchas in β=0.1. In β=0.26 with relatively low peak-to-peak amplitude (FIG.8, mode (ii)), the POD mode is similar to the fundamental mode of thelinearized Euler-Bernoulli beam equation, ρ_(s)h∂_(t) ² y+B∂_(x) ⁴y=0with free leading-edge and clamped trailing-edge boundary conditions.When the amplitude saturates with decreasing β, the POD mode deviatesfrom the fundamental mode, and the mode shape becomes more flattened.The dominance of the first POD mode in the decoupling of spatial andtemporal components of y(s,t) clearly shows that the oscillation of theflag sheet is in a stationary wave form instead of a traveling waveobserved in the general or standard (i.e., non-inverted) flag.

The first POD mode of the orientation angle θ, the angle between theinverted flag sheet and the −x axis at a given s/L (see, FIG. 6A), isalso presented in FIG. 11B. The value of θ increases linearly from theclamped trailing edge, but the slope of θ (curvature) reduces graduallyalong s and reaches to zero at the tip. In contrast to the mode shape ofy, the mode shapes of θ are almost identical in the flapping moderegardless of bending stiffness β.

In low mass ratio of O(10⁻³), the flag sheet can also flap with A/L>0.2in the range of 0.2<β<0.4 (FIGS. 12A and 12B), which is similar to theflapping range of the high mass ratio (FIGS. 7A and 7B). However, due tohigh fluid density and resultant high added-mass effects, theoscillation of A/L>1 occurs in a smaller range of β, 0.2<β<0.25. In manycases of the flapping mode, the flag sheet does not cross the y=0 lineafter rebounding, but repeats to bend and rebound in one y-side only soboth maximum and minimum y have the same sign. (Notably, this can occurin air as well, see example in FIG. 10 for 2 inch at 5.5 ms⁻¹ how theflag flaps only to one side.) For this reason (in the O(10⁻³) case),both A/L and fA/U decrease from those of high mass ratio (i.e., whencomparing FIGS. 7A/7B against FIGS. 12A/12B).

The periodic flapping in the low mass ratio is one of the interestingfeatures of the inverted flag. In the general flag, the critical bendingstiffness is nearly linear with μ for μ<1, and less than 10⁻⁴ at μ=10⁻³.However, in the inverted flag, the bending stiffness range of theflapping mode is not significantly changed by the mass ratio betweenO(1) and O(10⁻³). Only dynamical behaviors such as amplitude andflapping frequency are significantly affected.

In that the inertia effects of the sheet and the surrounding fluid arenegligible, static divergence instability rather than flutterinstability should be responsible for the unstable motion of thestraight sheet. As the lift force exceeds the restoring bending force indecreasing β, the straight sheet starts to bend in one side by bucklingand unsteady flows eventually induces limit-cycle flapping as apost-divergence behavior.

Once the sheet starts to deflect with high amplitude, the drag exertedon the sheet becomes a major source for bending. In this case, thedependence of aerodynamic drag and elastic strain energy on thedynamical modes may be examined. Only the cases of high mass-ratio O(1)are considered below. Drag coefficient C_(D) and non-dimensional elasticstrain energy E_(S) are defined as follows:

$\begin{matrix}{{{{C_{D}(t)} = \frac{D(t)}{\frac{1}{2}\rho_{f}U^{2}{HL}}}{and}{E_{S}(t)} = {\frac{\int_{0}^{L}{\frac{1}{2}{{BK}( {s,t} )}^{2}\ {s}}}{\rho_{f}U^{2}L^{2}} = {\int_{0}^{1}{\frac{1}{2}\beta \; {\kappa ( {s,t} )}^{2}\ {( {s/L} )}}}}},} & (2)\end{matrix}$

where K(s,t) is a dimensional curvature at a given curvilinearcoordinate s and κ=KL. Since fA/U is 0.1 to 0.2 in the flapping mode,unsteady transient force should not be neglected, especially when thesheet is in a high angle of attack against the free stream. Thus, themax C_(D) of the flapping mode, which occurs when the tip reaches at max|y|, is two to three times higher than the mean C_(D) of the deflectedmode (see FIG. 13A). Furthermore, although the sheet bends and relaxesrepeatedly, the mean C_(D) of a cycle in the flapping mode is comparableto that of the deflected mode.

The curvature κ of the clamped trailing edge at a maximally deformedphase increases from 4 up to 7 to 8 as β decreases from 0.3 to 0.1 inthe flapping mode. Meanwhile, κ of the trailing edge is between 5 and 8in the deflected mode. Because of high curvature comparable to thedeflected mode, the max E_(S) of the flapping mode ≈0.40 to 0.60 is muchlarger than the mean E_(S) of the deflected mode ≈0.15 (see FIG. 13B).As demonstrated comparing modes (iii) and (iv) in FIG. 8, despite lowerwind velocity, an elastic flag sheet of the flapping mode can benddownstream more and have higher dimensional strain energy than a sheetof the deflected mode. Moreover, the flapping mode generates the meanE_(S) of a cycle higher than that of the deflected mode.

The total kinetic energy of the incoming flow passing through themaximum frontal area during a bending phase is Ê_(K)^(d)=½ρ_(f)U³|ŷ|H{circumflex over (T)}. And |ŷ|H is the maximum frontalarea of the flag sheet during bending in either positive or negativey-side. {circumflex over (T)} is the time from when the tip of the sheetcrosses the y=0 line to when the sheet is at maximum deformation. From(2), the conversion ratio from fluid kinetic energy to strain energyduring bending is defined as:

$\begin{matrix}{{R = {\frac{{\hat{E}}_{S}^{d}}{{\hat{E}}_{K}^{d}} = \frac{\int_{0}^{L}{\frac{1}{2}B{\hat{K}}^{2}\ {s}}}{\frac{1}{2}\rho_{f}U^{3}{\hat{y}}\hat{T}}}},} & (3)\end{matrix}$

where Ê_(S) ^(d) is the dimensional strain energy at maximumdeformation. The conversion ratio R is between 0.2 and 0.4 in thebending phase of the flapping mode, and has a peak value near β=0.17 to0.20 (see FIG. 13C). Here, R indicates how much the elastic strainenergy can be stored from fluid kinetic energy flux for the possibleenergy transfer to other dissipative energy forms (e.g., fluid pumpingand electrical conversion per above, for piezo-based conversion, etc.).

Regarding the underlying physics, fluid flow visualization shows strongcorrelation between the development of vortex structures and theflapping dynamics of the inverted flag sheet (see FIG. 14). After thesheet 90 crosses the y=0 line, a vortex starts to form at the leadingedge. The leading-edge vortex continues to grow as the sheet bends (i)to (ii), and separates from the leading edge (iii) and moves downstream(iv) as the sheet rebounds and repeats on opposite side (v) to (vi). Asa result, alternately signed leading-edge vortices shed periodicallyinto the wake in response to the flapping frequency of the sheet. And(per above) even though the visualization in the water tunnel was onlyperformed in this study for the low mass ratio O(10⁻³), it is reasonableto infer that the flapping of the sheet and the shedding of theleading-edge vortices in the high mass ratio O(1) will also besynchronized.

Thus, the strain energy is stored while the leading-edge vortex is beingdeveloped during bending and reaches maximum when the vortex is fullydeveloped. The strain energy is released when the vortex starts toseparate from the leading edge and the sheet rebounds. From thisobservation, how optimal leading-edge vortex formation process isrelated with the magnitude of strain energy conversion may be studied.

For this purpose, the concept of Formation number is employed. Theformation number was first suggested in Gharib, M., Rambod, E. &Shariff, K. 1998, “A universal time scale for vortex ring formation,” J.Fluid Mech. 360, 121-140. For present purposes, Formation Number (F) isdefined as:

$\begin{matrix}{F = {\int_{t_{i}}^{t_{f}}{\frac{U + {u_{x}(t)}}{\overset{\sim}{y}}\ {{t}.}}}} & (4)\end{matrix}$

In the present system, t=t_(i) is when the tip crosses the y=0 line,t=t_(f) is when the flag sheet is at maximum deformation, |{tilde over(y)} is the y-coordinate magnitude of the tip at t=t_(f) and u_(x)(t) isthe x-directional velocity of the tip. The growth rate of the vortexstrength is reduced by the downstream motion u_(x) of the flag sheetwith a magnitude that can be up to 0.6 U. Therefore, u_(x)(t) isincluded in the definition of F.

In both low and high mass ratios, the formation number F of mostflapping cases is within a narrow range between 4.0 and 6.0 even thoughthe flapping dynamics are different among various conditions of β and μ(see FIG. 15). The F curve shown as a function of β is in a concaveshape and has a trough around β=0.2. It is found that F is in the rangeof 4.0 to 4.5 at β=1.7 to 2.0 where the energy conversion ratio is high(FIG. 13C).

Notably, this number is close to the formation number for efficientperformance of powered systems such as jet propulsion. The subjectanalysis further suggests that the formation number may also be used asa parameter to characterize the relation between optimal vortexformation and efficient storage of strain energy during bending in theself-excited flapping system.

Vertical Axis Flag Turbine

Historically, wind energy was first captured using vertical axis windturbines (VAWTs). While striving for higher conversion efficiencies,VAWT technologies were abandoned in favor of horizontal axis windturbines (HAWTs) at the cost of simplicity and durability. Meanwhile,VAWT farms are proving to generate more energy per unit land area thanHAWTs due to spacing requirements. In this section, features aredescribed suited to offer improve durability and elegance in VAWTdesign.

A VAWT turbine was built and tested with both rigid and compliantblades. The model was unable to sustain rotation when it was equippedwith rigid blades. In contrast, when the model was equipped withcompliant blades the turbine not only demonstrated remarkableself-starting capabilities but also produced power in connection withthe type of inverted flag instability and reconfiguration discussed anddemonstrated above.

VAWTs have several benefits compared to the modern horizontal axis windturbines (HAWTs) including directional independence from the wind,easier maintenance, reduced noise and visual signatures and ability toachieve closer spacing resulting in increased power density. Thesebenefits are caused by the way VAWTs differ from their horizontalcounterparts.

HAWTs are directionally independent from the wind. While this istypically viewed as a positive trait, the turbines have no mechanism bywhich they may turn out of the wind (like HAWTs) to prevent damage inhigh wind conditions. Designs to mitigate damages would be advantageous.Directional independence also necessitates a tricky design where theblades of a VAWT have higher forces on one side of the turbine incomparison to the other—in other words, a force differential orasymmetry across the turbine is required for operation. Statedotherwise, for VAWTs in use, one side of the turbine is moving upstreamin the wind and one side is moving downstream. Fundamentally, VAWTs spindue to a drag differential across the opposite sides of the turbine. Assuch, blade design is critical in determining VAWT performance.

This work explores using the mechanism of flexibility and associated(part-time) inverted flag dynamics (for flapping and/or reconfiguration)as well as general flapping flag type instability as pertinent to theissues of design durability and/or enhanced force differential across awind turbine.

In these designs, flexible blade react to forces (such as from wind)differently than rigid blades. For example, rigid blades will have adrag force that is always proportional to the square of velocity.However, the flexible blades can both enhance the drag force as well asdiminish the drag force. They can be in predetermined configurations toimprove performance (either actively or passively). They can naturallyor artificially flap or oscillate to help propel the turbine going withor against the wind.

In one aspect, the blades are able to reconfigure in response to highwind as per an example of a leaf that flips back and over, or moretechnically such as shown in mode/configuration (iv) in FIG. 8 andbeyond to a trailing free edge configuration. In another aspect, theVAWT includes at least one pair of flag type panels or sheets operatingin accordance with aspects described above and as further elaboratedupon below.

To characterize such performance, a flag-type VAWT model was built suchthat blades of different materials could easily be interchanged and theblade pitch angle set as desired. The turbine was studied experimentallyunder two conditions. In the first (hereafter referred to as “statictesting”), the turbine was prevented from rotating via a clampingmechanism. The clamping mechanism was constructed such that the turbinecould be clamped at several different angles relative to the free-streamflow. In the second condition, (hereafter referred to as “dynamictesting”), the turbine was free to rotate. A motor was applied in theopposite direction of the turbine's rotation in order to adjust theturbine's power output. In both situations, images of the blade behaviorwere recorded and analyzed in conjunction with the torque (in the caseof static testing) or power (in the case of dynamic testing) data.

Experimental results from static tests show the flexible flag-typeblades enhance the torque differential experienced across the turbine.Experimental results from dynamic tests show the turbine was unable torotate when otherwise identically-configured rigid blades were affixedto the turbine.

With the flexible blades, different modes of blade behavior resulted inturbine rotation. In addition, the turbine affixed with flexible bladeswas able to self-start from rest when wind velocity was raised.Analytical work was performed to help explain how the various modes ofblade behavior observed aided in turbine rotation. The characteristicsof the flexible blades are described in detail along with an analysis ofturbine performance.

A small scale VAWT was built and installed in a wind tunnel. Viewed fromabove, the turbine was configured as shown in FIG. 16A. Specifically,turbine 200 includes support arm(s) 202 turning on an axis 204. Thesupports secure flag type sheets (alternatively referred to as panels orblades in this section) 206, 208 opposite one another. As the turbineturns between the states shown in FIGS. 16A and 16B, upper blade 206,oriented as an inverted flag (90), bends (either passively or actively)such that it experiences higher forces than lower blade 208, oriented asa standard or general flag, that also bends (again, actively orpassively) such that it experiences lower forces. FIG. 16B furtherannotates the same structure in which α is the angle of attack of theblades, θ is the angle of the turbine's rotation and Φ is defined as theangle of attack of the blades to the free stream U_(∞).

In greater detail, FIG. 16 illustrates the turbine as-built, including avertical shaft 210 held between two bearings 212, 214 and secured in awind tunnel 220. The top bearing 212 was secured to the top 222 of thewind tunnel and the shaft passed through the floor 224 of the windtunnel where it connected to the bottom bearing 214. The lower end ofthe shaft passed through the bottom bearing to connect to one side of aFutek TRS605 torque meter and encoder 226 (alternativelyrepresenting/represented by a generator). The second side of the torquemeter and encoder system was connected to either a clamp 228 or a loaddepending on the experiment as further described below.

The support arms 202 were attached to shaft 210 at 180° apart. Betweenthem, blades 206 and 208 were secured between two optical anglers 230such that the blade angle (β) could be accurately set for each blade.

Different blade materials were selected to illustrate behavioraldifferences between blades of contrasting compliances per Table 2 below:

Young's Modulus Thickness Ca Material (E) (t) U_(∞) = 8:5 m/s Aluminum 69 GPa 0.003175 m (⅛ inch) 0.005 DELRIN 2.4 GPa 0.0015875 m ( 1/16inch) 1.13 DELRIN 2.4 GPa 0.00079375 m ( 1/32 inch)  9.04 Rubber   2 MPa0.003175 m (⅛ inch) 157The blades were mounted between arms 202 at a distance of 9 inches fromvertical shaft 210. Wind speeds in the tunnel 220 were varied between 5and 8.5 ms⁻¹.

For static testing, the turbine was fixed at various angles of rotationin order to focus on the force differential between the upstream anddownstream blades given the different parameters involved. The lowerside of torque meter and encoder was clamped to prevent rotation of theturbine while allowing for the data collection of the torquedifferential (τ_(c)) at each of the fixed angles. The torquedifferential is described by:

τ_(c)=(F _(U) −F _(D))R sin φ  (5)

where F_(U) and F_(D) are the forces acting on the upstream anddownstream blades respectively, R is the length of the moment arm andthe sin φ term is a geometric artifact stemming from the changingturbine angle relative to the free stream (U_(∞)).

Static testing was performed on each blade type at 15 different anglesevery 24° around a full 360° turbine rotation. Measurements wereconsolidated through superposition to every 12° over half of the fullrotation (e.g., data taken at a static angle of 192° is displayed at12°). In addition to the torque data, a high speed camera was mountedabove the tunnel and recorded images of the blades at a rate of 100frames/second during each of the experimental runs. These images enabledan analysis of the blade deformation observed in tests with thecompliant blades.

Dynamic tests were performed in which the lower side of the torque meterand encoder was connected to a DC motor mimicking the load a generatorwould apply on a turbine. The load was varied by discrete incrementsbetween the freely rotating situation with no load (τ=0) to the pointwhere the load was sufficiently high to prevent turbine rotation (φ=0).This load variation technique established the power curve and pinpointedthe regime where the turbine operated most efficiently.

Torque and angular velocity data were collected for every load caseafter the turbine was allowed to reach steady state operatingconditions. Turbine efficiency, as defined in equation 6, was calculatedand compared for each test.

$\begin{matrix}{\eta = \frac{\tau \; \varphi}{\frac{1}{2}\rho_{air}A_{swept}U_{\infty}^{3}}} & (6)\end{matrix}$

Tip-speed ratio, defined as the velocity ratios between the blade tip ascompared to the flow speed is described in equation 7.

$\begin{matrix}{\lambda = \frac{2\; \pi \; R\; \varphi}{U_{\infty}}} & (7)\end{matrix}$

Power curves expressing efficiency as a function of tip speed ratio weremade for both instantaneous values of torque and angular velocity aswell as values averaged over individual rotations.

In both static and dynamic tests, blade behavior was observed through ahigh speed camera mounted above the wind tunnel. In the case of therigid aluminum blades, no dynamic behavior was observed. However, in thecase of the flexible blades several types of behavior—flutter,reconfiguration and divergence—were observed. The specific type ofbehavior depended primarily on blade type and angle of attack.

These regimes are presented in FIG. 18. And while FIG. 18 delineatesthese regimes, their exact boundaries are neither constant across bladetypes nor across wind speeds. Nor does every regime of behavior existunder all conditions studied; further discussion is provided below.

Regarding the static testing, FIG. 19A shows static torque data for theturbine with rigid blades at several points around the turbine'srotation. The horizontal lines represent the average torque valuesacross the full turbine rotation. The average torque for the turbinewith rigid blades is near-zero for each set of velocity measurement,indicating that the turbine will not be able to sustain rotation duringdynamic testing in this case. Stated otherwise, while the magnitude oftorque increases with increased velocity there is no qualitativedifference in the averages.

Notably, the same qualitative similarities were found for all turbinesetups, therefore all further comparisons between turbine types will bemade at a free-stream velocity of 8.5 ms⁻¹. Also notable is the factthat when the upstream blade passes the downstream blade (effectivelyblocking the downstream blade from the wind, shortly after 90°) theturbine produces back-torque regardless of which blades are in use.While this is not desirable and would reduce the speed of rotationduring regular operation this trend is quite common in VAWTs.

What is most interesting to note is that in comparing the results inFIG. 19A vs. those in FIG. 19B (presenting flexible-blade data) is thatwhen the turbine exhibits positive torque, the flexible blades greatlyenhance the magnitude of torque exerted on the turbine. Moreover, in thesituations where the turbine is producing back-torque, the flexibleblades also reduce the negative impact and in some situations evencreate a positive (albeit low) torque in comparison to the turbine withrigid blades. Accordingly, examples of the flexible blade turbine have apositive average torque, indicating that the turbine will sustainrotation, even with a few moments of back-torque.

The shaded coding is also interesting to observe in FIG. 19B.Specifically, the graph is gray-scale coded to match the graphic in FIG.18. Turbine rotation in the darker grey region was associated with theblade undergoing large amplitude inverted-flag type flapping. This bladebehavior is associated with an increase in drag. Similar to the bladebehavior, the magnitude of drag was also periodic.

It was also observed that during these large amplitude flapping eventsthat the connections between the turbine shaft, torque meter and clampcreated a spring-like effect. Along these lines, every time the flappingblade produced substantial amount of force against these connections,they stored the energy. As the flapping blade released the force, theconnections discharged the stored energy. This resulted in largeamplitude oscillations of the torque data, matching the flappingperiodicity of the blade, and resulting in the large variations inamplitude observed in FIG. 19A in the case of the 1/32 inch DELRINblades (Ca≈10). This large amplitude flapping was also observed in thedynamic tests of this blade type—primarily when the turbine had littleor no rotational velocity either during startup or under heavy loading.

Also, even though they were not suitable/susceptible to the beneficialflapping in an inverted state, high Cauchy number (Ca≈100) blades (e.g.,the rubber blades tested) experienced high degrees of blade deformation,creating blades with curvature similar to those seen in a Savonius windturbine. Thus, they experienced a net positive average torque and may beuseful as compared to rigid blade counterparts, perhaps especially asfar as avoiding damage in high wind (or other flow) conditions isconcerned.

Other unique behavior associated with the testing is noted in thelighter grey regime in FIGS. 18 and 19B. Here, the blades exhibitedsmall amplitude flutter-type oscillations, similar to the flapping of astandard or general flag.

As for the dynamic testing (as suggested in the static testing), it wasfound that the turbine with rigid blades was unable to sustain rotation,even when no load was applied to the turbine and the turbine wasmanually started. In contrast, the turbines with flexible blades wereable to rotate and produce power.

As one example, FIGS. 20A and 20B show the power curve for the turbinesetup with 1/32 inch thick DELRIN blades. FIG. 20A shows instantaneousefficiency as a function of angular velocity and FIG. 20B the efficiencyfor one rotation of the turbine as a function of tip-speed ratio. Assuch, it is evident that variation in the instantaneous power curve is aresult of the unsteady angular velocity throughout one rotation. Thisvariation can be mitigated by methods such as using more than two bladesin the turbine configuration or setup.

Also notable, is that unlike other turbines where the power curve cutsoff well before the turbine reaches a zero angular velocity, thisparticular turbine power curve stretches into negative angular velocity.When a sufficiently high load is applied to the turbine, the turbinewill start to rotate in reverse. Reverse rotation was observed up untilthe point that the turbine's position allowed for blade flapping tocommence. The turbine would react to the impulsive force resulting fromthe flapping, and then rotate forwards. Such action clearly demonstratesa self-starting ability unlike any other VAWT. As such, while the regimein which the turbine rotates in reverse does not generate power, testresults indicate that it presents a solution to the standard start-upproblems of typical VAWT designs. In addition, it is hypothesized thatthrough appropriate blade distribution on a full sized turbine model,that this reverse rotation component can be eliminated.

FIGS. 21A and 21B show power data for two turbine loading conditions(namely 21A had a much higher load applied, and thus higher torqueoutput at a lower rotational speed and 21B has no load—other thaninternal friction) filtered through a low pass filter to better enablevisualizing overall trends. While the main periodicity corresponds toone full turbine rotation, it is interested that there is a localmaximum and local minimum in every half period of the power data.

While it was assumed the turbine is symmetric across any vertical planepassing through the rotational axis in the static tests, here it appearsthat the turbine is not symmetric. However, the dynamics of the systemcome into play and could also account for the asymmetry observed in thepower data. At low loads, the magnitude of this asymmetry (FIG. 21B) ismuch smaller suggesting that there are slight differences between thetwo blades of the turbine. As the load is increased on the turbine (FIG.21A), the asymmetry increases in magnitude dramatically. To understandthis behavior it is also important to understand the behavior of theblades. Blade behavior is highly dependent on the relative velocity ofthe blades. In addition, the blade behavior is directly connected to theextent of the power output.

In any case, the impact of the above-referenced flapping behavior onpower generation is clearly visible in the power data. Specifically,FIGS. 22A-22H display turbine blade position corresponding to the poweroutput data illustrated in FIGS. 23A and 23B. In the photographs, thewind flow moves from left to right, and the turbine rotatescounter-clockwise. The top blade in the images is moving against thewind and will be referred to as the upwind blade, while the blade in thebottom of the picture is moving with the wind and will be referred to asthe downwind blade.

As observed, an increase in power follows each flapping event. Powerdecreases immediately when the front blade of the turbine starts to passin front of the rear blade. Power increases as the wind “catches” thefront blade on its downwind pass. Indeed, through quantitativeobservation it is apparent that power production is at a minimum justprior to the wind catching the blade. Peak power production ends as theupstream blade passes in front of the downstream blade.

Time point A (pictured in FIG. 22A and indicate in FIG. 23B) correspondsto the point of lowest power production which occurs every halfrotation. This occurs just before the downwind blade catches the wind,resulting in the turbine picking up speed and once again producingpower. Between time points B and C (each shown in FIGS. 22B and 22C,respectively), the downwind blade is bending away from the wind in aprocess known as reconfiguration. This leads to a decrease in poweroutput from the turbine. After time point C (FIG. 22C) the bladereverses direction in its flapping, increasing the relative velocityseen by the blade and enhancing power output. This trend happens againwith times points D and E (FIGS. 22D and 22E) and once again with timepoints F and G (FIGS. 22F and 22G). Time point H (FIG. 22H) correspondsto the point where the upwind and downwind blades switch, with thepreviously upwind blade becoming the downwind blade and thus the powerproducing blade.

Such large amplitude flapping does not occur at all loading and/or bladecases. Flapping occurs as the loading on the turbine increases and theturbine progresses towards stall. Due to the turbines rotation, therelative wind velocity as seen by the blades is smaller in magnitudethat the free stream. Prior to the turbine approaching stall, therelative flow velocity has not yet reached the critical point necessaryto excite large scale flapping. However, of the loading cases whereflapping is exhibited, the behavior described in the flowchart of FIG.24 is representative.

As for dynamic startup, this activity was tested using the wind tunnelunder conditions starting from rest and quickly moving to a wind tunnelspeed of 8.5 ms⁻¹. As the wind velocity increased, the turbine rotatedinto a position of least resistance—typically with one blade in front ofthe rear blade. This position is similar to that in FIG. 22G.

In this configuration, the rear blade was seen to start flapping. Theflapping propulsion was then sufficient to propel the turbine such thatthe previously forward blade catches the wind and starts large-amplitudeflapping. And as shown under steady operation, this flapping instabilityproduced more rotational motion. As a result, each blade consecutivelywent through a large scale flapping event, resulting in increased poweroutput until the turbine reached steady state operating conditions.

In addition to the testing above, wind tunnel performance of the blades(of various flexibility) was compared to an analytical model. Existingmodels provided the theoretical framework to calculate the forces thatthe subject deformed turbine blades might produce. (See, FrédérickGosselin, Emmanuel de Langre, B. A. M. A., 2010, “Drag reduction offlexible plates by reconfiguration,” Journal of Fluid Mechanics 650,1-23; Luhar, M., Nepf, H. M., 2011, “Flow-induced reconfiguration ofbuoyant and flexible aquatic vegetation,” Journal of Limnology andOceanography, 2003-2017) where the Euler-Bernoulli beam equation forlarge deformations,

$\begin{matrix}{{M = {{EI}\frac{\partial\varphi}{\partial S}}},} & (8)\end{matrix}$

was worked through in length to yield:

$\begin{matrix}{{{EI}\frac{\partial^{3}\varphi}{\partial S^{3}}} = {\frac{1}{2}\rho_{air}C_{D}^{R}{AU}_{\infty}^{2}}} & (9)\end{matrix}$

with drag force as a function of the density of the fluid around theobject (ρ_(air)), the drag coefficient of the rigid blade whenperpendicular to the flow (C_(D) ^(R)), the cross-sectional area (asseen by the fluid flow), and the velocity of the fluid (U_(∞)) whichequation was then non-dimensionalized by introducing dimensionlessvariables and solved numerically.

However elegant, the model had a number of limitations. First, it onlyconcerned reconfiguration of the blades in the static situation. Second,the model only calculated the drag force and neglected lift created bythe turbine blades.

Nevertheless, through use of known geometry, torque on the turbine wascalculated and compared to measured quantities. The analytical model wasused to predict the forces acting on each individual turbine blade. Thebalance between the drag force of the two turbine blades was used topredict the drag force and the reconfiguration of the on both turbineblades. Once the drag force had been predicted, the forces are appliedto equation (5) to calculate the torque on the turbine at a givenstatically held position.

The model used model relied on the Cauchy number, a ratio between theinertial and elastic forces experienced by the blades. The exact form ofthe Cauchy number used was defined as:

$\begin{matrix}{{Ca} = {\frac{1}{2}\frac{\rho_{air}C_{D}^{R}{bU}_{\infty}^{2}L^{3}}{EI}}} & (10)\end{matrix}$

where I is the area moment of inertia,

$\begin{matrix}{{I = \frac{{bt}^{3}}{12}},} & (11)\end{matrix}$

so equation (10) could be rewritten independent of blade height (b), as:

$\begin{matrix}{{Ca} = {\frac{6\; \rho_{air}C_{D}^{R}U_{\infty}^{2}L^{3}}{{Et}^{3}}.}} & (12)\end{matrix}$

For every static turbine position tested, the torque predicted by themodel and the torque recorded through the experiment were compared.These results are plotted in FIG. 25A (rigid aluminum, Ca=0.005) and inFIG. 25B (for rubber, Ca=158).

For the rigid blades (aluminum), the analytical model predicts near zerotorque, as would be expected due to the symmetry of the two rigidblades. While the experimental results average to zero net torque, themagnitude of values at every fixed angle was much larger than thosepredicted by the analytical model as shown in FIG. 25A. This is likelydue to lift and blockage effects not considered in the analyticalsolution.

By contrast, the most compliant blades tested (rubber) demonstrated thegreatest agreement between the analytical solution and experimental dataas shown in FIG. 25B. This is likely due lack of inverted flag flappingdynamics as previously commented upon—especially demonstrated with the1/32 inch DELRIN (Ca=10) example—given that the model also omitted sucheffects.

Table 3 provides further comparison between torque found in testing andthe predicted values:

Experimental Analytically Average Predicated Thickness Torque AverageTorque Material (t) (Nm) (Nm) Aluminum 0.003175 m (⅛ inch) 0.01730.00049 DELRIN 0.0015875 m ( 1/16 inch) 0.18 0.11 DELRIN 0.00079375 m (1/32 inch)  1.0 0.80 Rubber 0.003175 m (⅛ inch) 0.62 0.3

In addition, a graphical representation comparing predicted to actualperformance was prepared. FIGS. 26A-26C show predicted states for rubberblades (206, 208) in connection with support arm(s) 202 againstcorresponding physical structure in FIGS. 27A-27C (static wind tunnelresults for rubber blades). Near identical correspondence is shown aswould be expected in view of the agreement expressed in thecorresponding case represented FIG. 25B.

Overall, a comparison of analytical and test results for the variouscases shows that for the regions where drag is the prevailing force,there is very good agreement between the model and test values.Likewise, in areas where one would expect lift to contribute more to theoverall turbine performance, there is some discrepancy between the modeland the experimental results. As such, use of the model enablesisolating lift-based effects of the forces.

Moreover, the test results clearly show that flexible blades can be usedto break the symmetry created by the two blades of a VAWT, enhancingpositive performance and decreasing negative performance in comparisonto a turbine with rigid blades. While current technologies avoid the useof flexible materials due to potentially erratic behaviors, the subjectdisclosure demonstrates that in some situations these very behaviors canbe beneficial to performance. Static tests showed that flexible bladeswere able to produce rotation where rigid blades would not. Dynamictests demonstrated enhanced performance with flexible blade flapping andbenefits start-up conditions as well.

Pressure Tunable Flag Structures

Hydro-skeleton type structures that maintain their integrity due to aninternally pressurized cavity are contemplated for use in the variousembodiments described above. Approaches are contemplated to alter shapeand/or change material characteristics (e.g. compliance) of theirflag-type elements. Hydro-skeleton features can provide a fast-actingapproach for quick control over the interactions of a structure with itssurroundings.

By controlling flapping flag or wind turbine blade shape, it is possibleto change the aerodynamic forces (lift and drag) experienced by thesame. Using dissimilar materials around the internal cavity allows thegross shape of the blade to change or morph when the internal cavity ispressurized.

By changing the compliance of a flapping flag or wind turbine blade, itis possible to change how much the body deflects and/or resonates orflaps due to external forces acting thereon. When a compliant structurehas an internal cavity, changing the pressure of the internal cavity canmodify or otherwise regulate the compliance of the structure (i.e.,effectively changing β or Ca). In this situation the cavity does notneed to be made of dissimilar materials.

A directionally morphable structure is made with an internal chamberembedded between two materials with dissimilar material properties. Thisinternal chamber can be pressurized, applying a normal force along allof the chamber surfaces. Since the materials are dissimilar, they reactdifferently to the application of the pressure. This allows theconstruction to bend towards to less elastic material. Stated otherwise,pressurizing the fluid within such a construction causes elasticmaterial containing the fluid to expand or stretch. With differentmodulus material employed for opposing layers in such a device, adifferential in expansion causes the panel to bend away from the moreflexible side. The key is a difference in elasticity that changes theeffective neutral axis upon pressurization.

Suitable flag-type constructions are pictured in FIGS. 1A and 1B. Ineach construction, opposite layers 2, 4 may comprise different materialssuch as ECOFLEX and PDMS (each being a common thermoplastic material)that are heat-bonded together. Various other materials and constructiontechniques including ultrasonic welding, use of adhesives, rivets, etc.may be employed in construction so long as pressure-tight chamber(s) areformed. As such, in another example, the morphing structures maycomprise a metal foil can be used in combination with PDMS or anotherpolymer. Of course, compliance control does not necessitate the use ofdissimilar materials in construction.

For either application (compliance and/or shape control) tubes orchannels may be produced on the micro-scale (i.e., at the scale of 1 mm,0.1 mm or less in cross-section diameter) within the device.Accordingly, such “micro-hydraulics” can allow for changing distributionof rigidity as well as shape. Indeed, the inclusion of multiple layersof micro-hydraulics (not show) will allow for shape change in differentdirections depending on the configuration of tubes in each layer.Alternatively, multiple layers may be used such that one set of chambersor channels is used for energy harvesting and another set is forcompliance and/or shape control. For example, one device could beflapping to produce a pressure differential. This pressure then could beused to force shape morphing or modulation of material properties inanother blade.

In any case (used in different fashion than described above), a systemlike that shown in FIG. 2 may be employed for hydraulic control. Inwhich case, with the check valves are omitted from the system, andelement 60 may operate as a pump instead of as a generator to controlflag shape and/or compliance while flapping. Otherwise pump 60 mayoperate in connection with feedback control circuitry 66 to moderatepressure in real time to control flag shape and/or compliancesimultaneous or in tandem with energy harvesting.

However the approach is implemented, FIGS. 28A-28F demonstrative bendingof a flag-type sheet or blade 90 thorough a progression of stages (90′,90″, etc.) due to internal pressure manipulation. As exemplified byFIGS. 1A and 1B, the flag chambers can have various geometries such asin those including a large reservoir 12 or a tube(s) or channels 22 ofvarious shapes. The shape and location of the chamber can be optimizedto improve performance. Internal chamber geometry may be used to definethe direction bending will occur and/or compliance is altered. Likewise,configurations can be made with single or multiple chambers (compareFIGS. 1A and 1D). Geometric differences can change the compliance and/ormorphing action experienced by the device.

The fluid that fills the chamber can be either liquid or gas. Choice offluid will depend on the application. Pressurization can be appliedrapidly and as a result the shape change can also happen rapidly.Devices can be used in all surrounding media (e.g. air, water, oil,etc.) with the same result of shape change due to the internalpressurization. Especially, by incorporation in the aboveconfigurations, the subject hydro-skeleton features can be used toenhance aerodynamic performance.

Variations

Methods of using these architectures are also within the scope of thisdescription. The subject methods may variously include assembly and/orinstallation activities associated with system use and product (e.g.,electricity) produced therefrom. Regarding any such methods, these maybe carried out in any order of the events which is logically possible,as well as any recited order of events.

Furthermore, where a range of values is provided, it is understood thatevery intervening value, between the upper and lower limit of that rangeand any other stated or intervening value in the stated range isencompassed and may be claimed. Likewise, while VAWTs with two bladeswere discussed above and may be claimed, this number is not exclusive.The subject turbines may include three or more blades. Also, it iscontemplated that any optional feature described herein may be set forthand claimed independently, or in combination with any one or more of thefeatures described herein.

Though embodiments have been described in reference to several examples,optionally incorporating various features, the present subject matter isnot to be limited to that which is described or indicated ascontemplated with respect to each variation. Changes may be made to thevariations described and equivalents (whether recited herein or notincluded for the sake of some brevity) may be substituted withoutdeparting from the true spirit and scope of the present subject matter.

Reference to a singular item includes the possibility that there are aplurality of the same items present. More specifically, as used hereinand in the appended claims, the singular forms “a,” “an,” “said,” and“the” include plural referents unless specifically stated otherwise. Inother words, use of the articles allow for “at least one” of the subjectitem in the description above as well as the claims below. It is furthernoted that the claims may be drafted to exclude any optional element. Assuch, this statement is intended to serve as antecedent basis for use ofsuch exclusive terminology as “solely,” “only” and the like inconnection with the recitation of claim elements, or use of a “negative”limitation.

Without the use of such exclusive terminology, the term “comprising” inthe claims shall allow for the inclusion of any additionalelement—irrespective of whether a given number of elements areenumerated in the claim, or the addition of a feature could be regardedas transforming the nature of an element set forth in the claims. Exceptas specifically defined herein, all technical and scientific terms usedherein are to be given as broad a commonly understood meaning aspossible while maintaining claim validity.

The breadth of the different embodiments or aspects described herein isnot to be limited to the examples provided and/or the subjectspecification, but rather only by the scope of the issued claimlanguage. It should be understood, that the description of specificexample embodiments is not intended to limit the scope of the claims tothe particular forms disclosed, but on the contrary, this patent is tocover all modifications and equivalents as illustrated, in part, by theappended claims.

What is claimed is:
 1. An apparatus comprising: a flag comprising atleast two opposing layers, the layers connected to one another to format least one chamber between the layers with at least one channel influid communication with the chamber; a mount, the flag secured to themount so that bending of the flag is required for movement in fluidflow; and a pump in fluid communication with the at least one channelfor pressurizing the chamber.
 2. The apparatus of claim 1, wherein theflag comprises input and output channels.
 3. The apparatus of claim 2,further comprising check valves in fluid communication with thechannels.
 4. The apparatus of claim 1, wherein the layers are made ofdifferent modulus materials.
 5. The apparatus of claim 1, wherein thelayers are made of the same material in different thicknesses.
 6. Theapparatus of claim 1, wherein the flag is secured to trail the mount asa standard flag setup in fluid flow.
 7. The apparatus of claim 1,wherein the flag is secured to lead the mount as an inverted flag setupin fluid flow.
 8. The apparatus of claim 1, wherein two of the flags aremounted to support arms to turn around an axis in fluid flow.
 9. Amethod comprising: flexing a flag in fluid flow to generate power; theflag comprising at least two opposing layers, the layers connected toone another to form at least one chamber between the layers; andaltering a property of the flag by pressurizing the chamber to enhancepower generation.
 10. The method of claim 9, wherein the flexing occursin a standard flag setup.
 11. The method of claim 9, wherein the flexingoccurs in an inverted flag setup.
 12. The method of claim 9, wherein theflexing occurs in a turbine setup.
 13. The method of claim 12, whereinthe turbine is a vertical axis wind turbine.
 14. The method of claim 9,wherein the property altered is stiffness.
 15. The method of claim 9,wherein the property altered is shape.
 16. The method of claim 15,wherein the shape altered is bend radius.
 17. The method of claim 9,wherein an array of flags are flexed.
 18. The method of claim 9, furthercomprising operating a pump to alter the flag property.
 19. The methodof claim 18, wherein the pump comprises a flag.
 20. The method of claim18, wherein the pump includes an electronic controller.